ACPAtmospheric Chemistry and PhysicsACPAtmos. Chem. Phys.1680-7324Copernicus PublicationsGöttingen, Germany10.5194/acp-16-8447-2016On the climatological probability of the vertical propagation of stationary planetary wavesKaramiKhalilkhalil.karami@kit.eduBraesickePeterSinnhuberMiriamhttps://orcid.org/0000-0002-3527-9051VersickStefanInstitute for Meteorology and Climate Research, Karlsruhe Institute of Technology, Karlsruhe, GermanySteinbuch Centre for Computing, Karlsruhe Institute of Technology, Karlsruhe, GermanyKhalil Karami (khalil.karami@kit.edu)12July20161613844784609October201517November20157June201612June2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://acp.copernicus.org/articles/16/8447/2016/acp-16-8447-2016.htmlThe full text article is available as a PDF file from https://acp.copernicus.org/articles/16/8447/2016/acp-16-8447-2016.pdf
We introduce a diagnostic tool to assess a climatological framework of the
optimal propagation conditions for stationary planetary waves. Analyzing 50
winters using NCEP/NCAR (National Center for Environmental
Prediction/National Center for Atmospheric Research) reanalysis data we
derive probability density functions (PDFs) of positive vertical wave number
as a function of zonal and meridional wave numbers. We contrast this quantity
with classical climatological means of the vertical wave number. Introducing
a membership value function (MVF) based on fuzzy logic, we objectively
generate a modified set of PDFs (mPDFs) and demonstrate their superior
performance compared to the climatological mean of vertical wave number and
the original PDFs. We argue that mPDFs allow an even better understanding of
how background conditions impact wave propagation in a climatological sense.
As expected, probabilities are decreasing with increasing zonal wave numbers.
In addition we discuss the meridional wave number dependency of the PDFs
which is usually neglected, highlighting the contribution of meridional wave
numbers 2 and 3 in the stratosphere. We also describe how mPDFs change in
response to strong vortex regime (SVR) and weak vortex regime (WVR)
conditions, with increased probabilities of the wave propagation during WVR
than SVR in the stratosphere. We conclude that the mPDFs are a convenient way
to summarize climatological information about planetary wave propagation in
reanalysis and climate model data.
Introduction
Climatology of the zonal mean zonal wind (left) in and the vertical
shear of zonal mean zonal wind (right) for the Northern Hemisphere during
DJF. The units are ms-1 for zonal mean zonal wind and
ms-1 km-1 for the vertical shear of zonal mean zonal wind
respectively.
The impact of the background atmospheric state on planetary wave propagation
was first investigated by based on linear wave
theory. They showed the importance of the background zonal wind for the
vertical propagation of large-scale waves from the troposphere into the
stratosphere. They found that vertical propagation of stationary planetary
waves can only occur when the zonal mean zonal wind is positive. In addition,
a strong stratospheric polar night jet of the Southern Hemisphere during
winter will block and possibly reflect large-scale waves. This implies that
the zonal mean zonal wind should be smaller than a critical value for
vertical propagation. This theory also suggests that large-scale waves (zonal
wave number =1, 2, 3) are more likely to propagate upwards because their
associated critical wind speeds are higher. Studies by ,
, , ,
, and not only confirmed this theory but
also stressed the importance of vertical shear of the zonal mean zonal wind
as well as the vertical gradient of the buoyancy frequency for vertical
propagation of large-scale waves.
introduced the refractive index for stationary planetary
waves (or alternatively vertical wave number) as a diagnostic tool for
studying the influence of the background zonal flow on planetary wave
propagation. According to linear wave theory planetary waves, away from the
source regions, tend to propagate toward the region of large positive
vertical wave number squared. The existence of Rossby waves is prohibited
where the vertical wave number squared is small or negative, which can happen
if the zonal mean zonal wind is easterly, or westerly exceeding the critical
wind speed.
The refractive index of Rossby waves as a diagnostic tool provides a
framework in which the dynamical forcing of the stratosphere by tropospheric
waves can be investigated. However, as shown by the
traditional analysis of the refractive index squared makes it difficult, if
not impossible, to study the climatological state of the background flow for
propagation of planetary waves. In calculating the climatology of the
refractive index squared, the problem arises from averaging a time series
that could consist of positive and negative values that may cancel each other
and hence makes the interpretation of climatologies of this quantity
difficult. Another weakness of the vertical wave number is that it is somewhat
vague. pointed out that, while using the vertical
wave number as a diagnostic tool one should not overemphasize the details,
since it is a qualitative guide. For instance found that
planetary waves can only propagate when and where the vertical wave number
squared is positive and very large or avoid the region of large negative
values of the vertical wave number. The vagueness arises from vague
expressions such as “very large positive” and “very large negative”
values of the vertical wave number which demonstrates the arbitrariness of the
classic time mean diagnostic.
Here we attempt to address the modeling of such vagueness which has not been
previously addressed. We present an algorithm based on fuzzy logic theory
which addresses the above-mentioned vagueness and provides an estimate of the
favorability of atmospheric background conditions for planetary wave
propagation as a function of latitude and altitude. Any diagnostic tool
should be consistent with the general knowledge about stationary Rossby wave
propagation condition (Table ). The first and second criterion
of Table are the most important findings of the seminal
papers of and . They made a
great contribution on the understanding of the propagation of planetary-scale
disturbances from the troposphere into the stratosphere.
based on the wave-mean flow interaction theorem
showed that the planetary waves also have a strong influence on the zonal
mean zonal wind. and argue that
only ultra-long waves (wave numbers 1–3) have the capability to propagate
from the troposphere into the middle atmosphere. Criterion 3 expresses
that the jet maxima block the planetary wave propagation and penetration
through the jet maxima is prohibited . The study of
shows that the key parameter that controls the
planetary wave propagation is the properties of the tropopause which acts
like a valve for the vertical wave propagation from the troposphere into the
stratosphere. Furthermore the study of and
indicated that the large positive vertical shear of zonal wind at the
tropopause height tends to enhance wave propagation (criterion 4).
and have discussed the importance of
vertical shear of zonal mean zonal wind on the vertical propagation of Rossby
waves. showed that penetration of planetary waves
from the troposphere into the stratosphere is sensitive to small changes in
the vertical shear of zonal wind near the tropopause height.
identified that a positive vertical shear of zonal wind enhances wave
propagation across the tropopause. Similarly large negative shear of zonal
wind tends to trap the planetary waves in the troposphere and hence less is
left to penetrate into the stratosphere. Any diagnostic tool that attempts to
provide a climatology of stationary Rossby wave propagation conditions should
reflect this theory. In fact, we try to develop an algorithm that is capable
of demonstrating the enhancing influence of positive vertical shear of zonal
wind and impeding influence of negative vertical shear of zonal wind on
stationary Rossby wave propagation from the troposphere to the stratosphere.
A summary of known facts about stationary Rossby wave propagation.
Any diagnostic tool that attempts to provide a climatology of stationary
Rossby wave propagation conditions should be consistent with these criteria.
These criteria refer only to the linear waves.
1For all stationary Rossby waves the most favorable propagation conditions are in the lower troposphere of the mid-latitude region. Upper troposphere and lowermost stratosphere of mid-latitude regions are also favorable for Rossby wave propagation. and 2For large-scale waves (horizontal and meridional wave numbers 1 to 3) the probability to propagate vertically is highest. and 3Rossby waves tend to propagate on the edges of strong westerly winds and avoid penetrating through the jet maxima. Therefore, the strong stratospheric polar night jet of the Southern Hemisphere in the winter will block and reflect large-scale waves.4Strong vertical shear (positive) is likely to enhance the vertical propagation of waves.
Figure shows the climatology of the zonal mean zonal wind and the
vertical shear of zonal mean zonal wind (ms-1 km-1) for the
Northern Hemisphere winter months. Northern Hemisphere winter months include
December, January and February (DJF) and Southern Hemisphere winter months
include June, July and August (JJA). Due to the larger meridional temperature
gradient between the tropics and mid-latitudes, the magnitude of the wind
shear between 20 and 40∘ N is about four times stronger than the
vertical shear at higher latitudes. Regardless of magnitude, it is evident
that it is positive in the troposphere and negative in the stratosphere in
this latitude band. The importance of the wind shear and buoyancy frequency
for the upward wave propagation is discussed by .
Data and method
In the current study we used daily mean zonal wind and temperature from the
National Center for Environmental Prediction-National Centre for Atmospheric
Research (NCEP-NCAR) to calculate the vertical wave number
of Rossby waves for 50 winters (1961–2010) of both Northern and Southern
hemispheres. The vertical wave number for stationary planetary waves is
defined as
mk,l2(y,z)=N2f2cos2(ϕ)qϕ‾u‾-ka2-πl2a2-fcos(ϕ)2NH2,
where
qϕ‾=cos(ϕ)2Ωacos(ϕ)-1a2∂∂ϕ∂∂ϕ(u‾cos(ϕ))cos(ϕ)-f2ρ0∂∂z(ρ0∂∂zu‾)N2
is the meridional gradient of the zonal mean potential vorticity which is a
fundamental quantity in Planetary wave dynamics and the stability of the
zonal mean flow . Here H, k, l, ρ0, f,
N2, a, Ω and ϕ are the scale height, zonal and meridional
wave numbers, air density, Coriolis parameter, buoyancy frequency, the Earth's
radius and rotation frequency and latitude respectively . The definition of the current version of the vertical
wave number of Rossby waves that depends on the two-dimensional wave numbers
(zonal and meridional wave numbers) can be found in .
Figures and show the time mean vertical wave number
(in the plots weighted with the Earth radius squared) of 50 winters for
Northern and Southern hemispheres respectively. The dependence of the time
mean vertical wave number on the zonal (k=1,2,3) and meridional wave numbers
(l=1,2,3) is visible in both figures. It can be seen that the multi-year
average of time mean vertical wave number gives unsatisfactory results. For
instance, for (k, l)=(1,1) very high values of the vertical wave number
squared are found in high latitudes of the troposphere and the lower
stratosphere. Moreover, in most areas of mid- and high latitudes of the
troposphere alternating positive and negative values of the vertical
wave number squared leads to a noisy structure and makes the interpretation
very difficult. The problem originates from overlapping of positive and
negative values in the time series and results in a reduction of
climatological information. Such features of the time mean vertical
wave number are also discussed by others . Too
high values of time mean vertical wave number northward of 75∘ N in
the lower stratosphere are not consistent with criterion 3 in
Table , because the strong jet is expected to block wave
penetration from the troposphere to the stratosphere. The time mean vertical
wave number is also not able to capture the meridional wave number dependency
on the wave propagation conditions (criterion 2 in Table ). For
example in the Southern Hemisphere, the difference between time mean for
wave (2,1), (2,2) and (2,3) in the stratosphere (above 100 hPa) is
small, suggesting no considerable influence from the meridional wave numbers
on the vertical propagation of planetary waves from the troposphere to the
stratosphere. In the current study, the time mean vertical wave number squared
is calculated by the time mean of the instantaneous vertical wave number
derived from the daily zonal mean field. As shown in Figs. and
the time mean vertical wave number has a noisy structure. One
possibility to reduce the noise level is to calculate the vertical wave number
of the time-mean zonal mean fields instead (Fig. ). However
time-dependent Rossby waves propagate on the instantaneous atmospheric state
and not on the time-averaged fields. Therefore we focus on an approach to
reduce the level of noise in the time-averaged instantaneous vertical
wave number.
Climatology of vertical wave number squared (a2mk,l2(y,z)) of
50 winters (1961–2010) in the Northern hemisphere. Regions with negative
a2mk,l2(y,z) are shaded in gray.
Climatology of vertical wave number squared (a2mk,l2(y,z)) of
50 winters (1961–2010) in the Southern Hemisphere. Regions with negative
a2mk,l2(y,z) are shaded in gray.
Probability of positive vertical wave number squared
introduced the frequency distribution of days with negative
vertical wave number squared as an alternative metric to describe how
planetary waves can propagate. Figure shows the probabilities of
positive vertical wave number squared for Northern Hemisphere winter time
expressed as the percentage of days with positive mk,l2(y,z) for wave
(1,1), (1,2) and (1,3). By comparing to the time mean of the same waves
we conclude that this quantity is capable of describing the required wave
properties better than the time mean of mk,l2(y,z). However, it results
in high values of probability between 20 and 40∘ N in the lower and
middle stratosphere. This might be an over-optimistic result, because it is
due to small positive values at these locations that exist throughout the
winter season. In this respect the climatology of probability of positive
refraction index squared does not meet the criterion 4 in
Table .
Further evidence to show the importance of ∂∂zu‾ for vertical propagation of Rossby waves can be provided by
calculating the normalized vertical component of the Eliassen-Palm (EP) flux.
Figure shows that the normalized vertical component of EP flux
has a minimum at the tropopause, indicating that upward penetration of waves
is suppressed by the negative values above tropopause heights as suggested by
. Sensitivity of mk,l2(y,z) to u‾ can be
studied by comparing the values of a2qϕ‾u‾ and a2qϕ‾10ms-1. Figure shows the
climatology of a2qϕ‾u‾ and a2qϕ‾10ms-1 for DJF in the Northern
Hemisphere. The subpolar maxima of a2qϕ‾u‾ in the troposphere are not related
to small values of the zonal wind at these regions, since by taking away the
u‾, the maxima are shifted to subtropics (25–40∘ N).
This implies that small values of u‾ rather than
∂∂zu‾ at subpolar regions cause the
maxima of mk,l2(y,z) at these regions.
Probability of positive
vertical wave number squared for Northern Hemisphere wintertime for wave
(1,1), (1,2) and (1,3).
Climatology of vertical component of EP flux normalized by vertical
component of EP flux at 850 hPa for DJF at Northern Hemisphere.
Discontinuity of this quantity at the tropopause heights indicates the strong
suppression of wave penetration from troposphere into the stratosphere at
lower stratosphere. discuss the importance of the
abrupt change of the buoyancy frequency at the tropopause level for the
suppression of the upward wave propagation.
climatology of a2qϕ‾u‾ (left)
and a2qϕ‾10 (right) for DJF in the Northern
Hemisphere.
Probability of favorable propagation condition for Rossby waves
A long standing issue in the interpretation of mk,l2(y,z) is its
vagueness. As suggested by , large waves tend to propagate
in regions of positive vertical wave number mk,l2(y,z) while they may be
trapped in vertical direction where mk,l2(y,z)<0. Here (in the light of
fuzzy sets and logic), we attempt to address the modeling of such vagueness.
Fuzzy logic is a mathematical method for answering questions with imprecise
information (such as very large or very small vertical wave number), it deals
with reasoning that is approximate rather than fixed and precise. The basic
approach is to assign a value between zero and one to describe the range
between the upper and lower limit. The upper and lower limits refer to the
maximum and minimum values of any variable. Within these limits fuzzy logic
assigns a membership value function (MVF) .
Here we assume that instead of each of the individual mk,l2(y,z,t)
contributing equally to the time-mean mk,l2(y,z), some
mk,l2(y,z,t) contribute more than others. In this way, we distinguish
between small positive and very large positive values to let very large
positive values influence the final result more than small positive values.
In this way classes or sets whose boundaries are not sharp will be
introduced. We introduce μRo(y,z,t) as the Rossby wave MVF which
provides mPDF and estimate the probability of favorable propagation
conditions of Rossby wave PrRo(y,z), as a function of latitude and height. We
also provide the physical basis of the proposed method. For a detailed
discussion of membership value function (MVF), see the Appendix A.
The advantage of our analysis over the traditional analysis of the vertical
wave number is that without any reduction in the information due to
cancellation of negative and positive values of the vertical wave number
squared, we estimate the likeliness for planetary waves to propagate from one
region to another at any time, altitude and latitude.
In Fig. the black curve shows the MVF used in the calculation
of favorable propagation condition of Rossby waves. For the negative
mk,l2(y,z,t) region (part a) this function suggests that the rate of
attenuation is very high and therefore wave propagation is prohibited in this
region. Since our method is still based upon the linear wave theory, we
assume a linear relationship between the magnitude of the mk,l2(y,z,t)
and the probability of favorable propagation conditions for positive
mk,l2(y,z,t) in a way that the higher the values of the
mk,l2(y,z,t) the chances of propagation for the Rossby waves increases
linearly (part b). Large values of the mk,l2(y,z,t) occur near the
critical line where zonal mean zonal wind approaches zero
(u‾<0.5 ms-1 in this study). This region is also not
favorable for Rossby wave propagation since at this region the linear wave
theory breaks down and waves start to break and the waves are absorbed
(part c). The region where vertical wave number squared is larger than 600 is
not favorable for wave propagation. At these regions the zonal mean zonal
wind approaches zero. This condition often happens in the upper
troposphere/lower stratosphere where westerlies become weak in the winter
season near the Arctic. Therefore most of the differences between
Figs. and for Rossby wave (1,1) at the
above-mentioned regions can be associated with setting μRo to
zero for mk,l2>600. In the study of the effect of the
critical line on Rossby wave propagation is neglected since all the positive
values of the mk,l2(y,z,t) are regarded as small and very large
positive values of the mk,l2(y,z,t) are equally favorable places for
wave propagation. In fact very high values of the mk,l2(y,z,t) are not
necessarily favorable conditions for the Rossby wave propagation. In this
study the mk,l2(y,z,t) higher than 600 is considered as the critical
line region, obtained from the climatology of the vertical wave number when
u‾<0.5 ms-1. As we will show, this function gives us an
improved picture of planetary wave propagation conditions in climatologies.
Higher values of PrRo(y,z) provide a window of opportunity for
planetary waves to propagate at any latitude and height. Likewise, smaller
values of PrRo(y,z) demonstrate the places where Rossby waves are
likely to be trapped in the vertical direction. The sensitivity of
PrRo(y,z) values to the shape of the MVF function is discussed in
Appendix A.
MVF used in the calculation of favorable propagation condition of
Rossby waves (black curve). Red lines show MVF for calculating probability of
positive vertical wave number which are used by . In their study
the effect of the critical layer (part c) is not considered.
Results and discussions
Probability of favorable propagation condition for Rossby waves
derived from 50 winters (1961–2010) in the Northern Hemisphere. The higher
the values, the more convenient it is for planetary waves to propagate to that
region. In contrast, planetary waves are likely to be trapped in the
vertical direction when the value of this quantity is small.
Figure demonstrates the climatology of probability of favorable
propagation conditions of Rossby waves for zonal wave numbers (k=1, 2, 3) and
meridional wave numbers (l=1, 2, 3) for the Northern Hemisphere winter
season. The most common feature for all waves is their rather large
probability to propagate in the troposphere (below 200 hPa) in winter
season. It is also evident that the most favorable propagation condition is
in the lower troposphere of the mid-latitude region. The values of
Fig. are independent of Rossby wave generation and explain how
the waves, when generated, would propagate given the structure of the mean
flow. However, the regions of highly favorable Rossby wave propagation and
source region for wave generation (asymmetries at the surface, land-sea
contrasts, and sea surface temperature asymmetries) are coincident. It is
also clear that longer waves have more opportunity to penetrate the
stratosphere.
by using ray tracing technique from geometrical
optics and wave propagation in a slowly varying medium, showed that wave rays
which are parallel to the group velocity vector tend to refract toward large
vertical wave number squared. They also found that Rossby waves have a
tendency to propagate along great circles and most of the upward propagation
of Rossby waves will be refracted toward the equator (even if the vertical
wave number squared were positive at all heights in their study). Similar to
this theory, we also found a channel or waveguide of large probability of
favorable propagation condition for Rossby waves. The strong westerlies act
as a waveguide of Rossby waves and direct them vertically through the
tropopause and allow them to penetrate to higher altitudes from their source
region (troposphere). These areas are south of 40∘ N in winter of
the Northern Hemisphere for large waves and are indicated by
PrRo(y,z)>50 %.
The study of also revealed that Rossby waves tend
to propagate on the edges of strong westerlies and avoid penetrating through
the jet. This fact is also clear in our results, where north of
60∘ N and above 200 hPa, the probability of favorable condition for
Rossby waves show relatively smaller values, compared to similar altitude
ranges between 30 and 50∘ N. The maxima south of 40∘ N at
100 hPa in the mPDF shows that the region is favorable for wave propagation.
At the same region, the vertical component of the EP fluxes have small
magnitudes. However as shown in the horizontal component of EP
fluxes has large values at this region (Fig. 5e in the study of
). Since the current study concentrates only on the vertical
wave propagation, not all aspects of Fig. can be directly
compared with Fig. . The same climatologies as
Fig. are presented in Fig. for the Southern
Hemisphere. Similar to the Northern Hemisphere, all large-scale waves have a
rather large chance to propagate in the troposphere in winter. It can be seen
that the larger the waves, the higher the probability of favorable conditions for them to
propagate upward.
The same as Fig. but for Southern Hemisphere
wintertime.
The differences between the probability of positive vertical
wave number squared and the probability of favorable propagation condition of
stationary Rossby waves.
Figure demonstrates the differences between probability of
positive vertical wave number (calculated by PDFs) and probability of
favorable propagation condition of Rossby waves (calculated by mPDFs) for
Northern Hemisphere wintertime for wave (1,1), (1,2) and (1,3). The
maximum difference is found at 20–40∘ N of the middle and upper
troposphere which can reach to 50 %. This unsatisfactory result of the
probability of positive vertical wave number is due to small positive values
at these places which is consistent throughout the winter season. The area of
maximum difference between PrRo(y,z) and probability of positive
vertical wave number remains the same for all wave numbers at both Northern and
Southern hemispheres (not shown).
As Figs. and show the most important difference
between the Northern and Southern hemisphere occurs in the high latitudes of
the stratosphere, where in the Northern Hemisphere, zonal wave number =1
has a good opportunity to propagate (PrRo(y,z)>40 %), while in
the Southern Hemisphere it has a rather poorer chance to propagate. This is
consistent with the theoretical explanation of the vertical propagation of
Rossby waves from the troposphere to the stratosphere by
. The zonal mean zonal wind should be weaker than a
critical strength for upward propagation of Rossby waves. The strong
stratospheric winter polar vortex of the Southern Hemisphere will block and
reflect wave activity. The critical strength depends on the scale of the wave
and is not a function of the background zonal regime.
A significant piece of information which is lost from the time mean of
mk,l2(y,z) is the role of meridional wave numbers on the wave
propagation conditions. For instance in the Southern Hemisphere, the
difference between the time mean of mk,l2(y,z) for wave (2,1),
(2,2) and (2,3) in the stratosphere (above 100 hPa) is not large which
is one of the unsatisfactory results of time mean of mk,l2(y,z). It is
only in the light of PrRo(y,z) values that we can understand the
impact of meridional wave numbers on the wave propagation in the stratosphere.
Note that, at the same latitude range of the Southern Hemisphere,
PrRo(y,z) values are as high as 45% for wave (2,1) in
mid-latitudes of stratosphere, while the PrRo(y,z) values reach to
less than 5 % for wave (2,3).
Usefulness and appropriateness of PrRo(y,z)
In order to test the appropriateness of the PrRo(y,z) in
climatological studies of stationary planetary wave propagation, we further
investigate the sensitivity of the PrRo(y,z) to different zonal
flow regimes in the stratosphere. Following , we
constructed two data sets based upon the strength of the westerlies in the
lower stratosphere (50 hPa) at 65∘ N. According to the
criterion, if the background flow is westerly and
smaller than the latitude and wave number dependent critical Rossby velocity,
the planetary waves can penetrate from the troposphere into the stratosphere,
otherwise wave reflection occurs and tropospheric flow may be modified.
strong vortex regime (SVR) is identified when
u‾50(65∘N)>20 ms-1 and weak vortex regime
(WVR) is considered when
0<u‾50(65∘N)<10 ms-1, where
u‾50(65∘N) is the 50 hPa zonal mean zonal wind at
65∘ N. The 20 ms-1 threshold reflects the critical Rossby
velocities (20 ms-1) for ZWN =1 for a climatological Northern
Hemisphere zonal wind profile. The WVR events do not correspond to the sudden
stratospheric warmings (SSWs) in the current study. Since during SSWs the
linear wave theory breaks down and waves start to break and the waves are
absorbed, the vertical wave number and probability of the favorable wave
propagation (both are based on the linear wave theory) have limitations for
studying the wave propagation during SSWs.
Periods of polar vortex regimes lasting for at least 30 consecutive days in DJF; left: strong vortex regime. Right: weak vortex regime.
Strong vortex regime (SVR) Weak vortex regime (WVR) Starting dateEnding dateStarting dateEnding date20 Dec 196120 Feb 196220 Dec 196827 Jan 196924 Dec 196328 Feb 196428 Dec 198413 Feb 19853 Jan 196728 Feb 19679 Dec 199811 Jan 19991 Dec 197528 Feb 19762 Jan 200428 Feb 20041 Dec 198714 Jan 198816 Dec 198817 Feb 198917 Dec 198928 Feb 19901 Dec 199118 Jan 19925 Dec 199211 Feb 19931 Dec 199418 Jan 19957 Dec 200421 Feb 200530 Dec 200626 Feb 200723 Dec 200713 Feb 2008
Table demonstrates the periods of different polar vortex
regimes that last for at least 30 consecutive days in DJF. Since in DJF the
stratospheric flow consists of strong westerlies (in the absence of vertical
wave propagation), the number of SVR events is higher than WVR events. The
results of mk,l2(y,z) and PrRo(y,z) for WVR and SVR for wave
(1,1) are presented in Fig. . It is found that in comparison to
climatologies (Fig. ) both WVR and SVR show similar patterns.
However, the waveguide at mid-latitudes is much narrower in SVR than WVR. In
addition, the average values of PrRo(y,z) in the stratosphere are
greater in WVR than SVR. These results show that planetary waves have more of
a chance to penetrate and force the stratosphere in WVR than SVR. In other
words, values of PrRo(y,z) are sensitive to stratospheric
westerlies and are consistent with the general knowledge about planetary wave
propagation from the troposphere to the stratosphere. An enhancement of wave
propagation northward of 70∘ N in the lower stratosphere and a
slight reduction in the favorability of wave propagation between
50 and 70∘ N in the stratosphere are found for WVR. On the other hand
it can be seen that due to the high level of noisiness the interpretation of
the difference of mk,l2(y,z) between WVR and SVR is very difficult.
Since the highest difference in the favorability of wave propagation between
WVR and SVR occurs northward of 50∘ N in the stratosphere, we
further calculate the difference in the vertical component of EP flux between
WVR and SVR in this region (Fig. ). An enhancement of vertical EP
flux is obtained northward of 65∘ N in the lower stratosphere during
WVR while a decrease in this quantity is obtained southward of this region in
the middle and upper stratosphere. By comparing the differences of
mk,l2(y,z), PrRo(y,z) and vertical component of EP flux
during WVR and SVR, it can be seen that the pattern of differences between
PrRo(y,z) and vertical component of EP flux are similar. Therefore,
based upon these analyses, we suggest that this diagnostic tool can be useful
for studying the propagating properties of the planetary waves.
a2mk,l2(y,z) (first row) and PrRo(y,z) (second
row) during WVR and SVR.
Same as Fig. but restricted to the 100–10 hPa range
for the vertical component of EP flux. The values are divided by 105.
Since the highest differences in the mk,l2(y,z) and PrRo(y,z)
between WVR and SVR are in the high latitude stratosphere the vertical
component of EP fluxes are shown in this region.
Conclusions
Climatological values of the time mean of the vertical wave number squared
derived from 50 winters (1961–2010) of both Northern and Southern
hemispheres are calculated to show several problematic features of this
important quantity in climatologies. In order to improve these unsatisfactory
results, we introduced probability density functions (PDFs) of positive
vertical wave number as a function of zonal and meridional wave numbers. We
also compared this quantity with a modified set of PDFs (mPDFs) and
demonstrated their superior performance compared to the climatological mean of
vertical wave number and the original PDFs. Without any reduction in the
information, PrRo(y,z) estimates the likeliness for stationary
Rossby waves to propagate from one region to another at any time, altitude
and latitude in a climatological sense. The higher the PrRo(y,z)
the easier it is for planetary waves to propagate. Smaller values of
PrRo(y,z) demonstrate the places where Rossby waves are likely to
be trapped in the vertical direction. It is also found that by using this
quantity one can easily study the difference in stationary Rossby wave
propagation between different meridional wave numbers without the difficulty
of the interpretation of the noisy structure of the time mean vertical
wave number. Our diagnostic tool is also capable of demonstrating the
enhancing influence of positive vertical shear of zonal wind and impeding
influence of negative vertical shear of zonal wind on stationary Rossby wave
propagation from the troposphere to the stratosphere. The better performance
of the mPDF suggests that relatively small but positive numbers of the
vertical wave number squared play an important role to offer a favorite
propagating condition for planetary waves in the stratosphere. This
diagnostic tool successfully shows that for WVR there is more space for the
vertical propagation of Rossby waves from the troposphere to the
stratosphere. In contrast, SVR tend to block and reflect vertical propagation
of stationary Rossby waves. It is also worthwhile mentioning that both the
vertical wave number and probability of the favorable wave propagation are
still qualitative tools to study the vertical propagation of Rossby waves
from the troposphere to the stratosphere. Since our diagnostic tool is
consistent with the theoretical understanding of vertical propagation of
Rossby waves from the troposphere to the stratosphere, we suggest that this
diagnostic tool has the capacity to be used in assessing planetary wave
propagation conditions in climate models.
Data availability
The NCEP/NCAR data set is publicly available at
http://www.esrl.noaa.gov/psd.
The probability of favorable propagation condition of Rossby waves
PrRo(y,z) can be written as
PrRo(y,z)=∑t=1nμRo(y,z,t)∑t=1nt×100,
where μRo(y,z,t) as modified set of PDFs (mPDFs) is defined as:
μRo=0ifmk,l2≤0,(8.3×10-4×mk,l2(y,z))+0.5if 0<mk,l2<600,0ifmk,l2≥600
Here 8.3×10-4 is the slope of line b in Fig. . The
variable t is the time step and in the current study the daily mean values of
the temperature and zonal wind are used in the calculations. In the study of
PDFs (red lines in the Fig. ) are defined as
μRo=0ifmk,l2<0,1ifmk,l2>0,
In order to test the sensitivity of PrRo(y,z) to the shape of MVF,
we evaluated the values of PrRo(y,z) for several potential MVFs.
Figure demonstrates the shapes of three MVFs that are used to
calculate the values of PrRo(y,z). It can be seen from
Fig. (first row) that MVF1 gives unsatisfactory results above
200 hPa, where for wave (3,3) we expect very low values of
PrRo(y,z) poleward of 40∘ N. This function (MVF1) neglects
the fact that Rossby waves tend to quickly attenuate in low values of
vertical wave number squared. The values of PrRo(y,z) can reach as
high as 50 % at these latitudes and altitudes. MVF2 and MVF3 also give
unrealistic results where the values of PrRo(y,z) are too low in
the stratosphere for all waves. These MVFs block all waves in the
troposphere. Furthermore, they do not provide any waveguides in which Rossby
waves can penetrate from troposphere to the stratosphere.
Shape of three MVFs that are used to calculate the values of
PrRo(y,z).
Probability of favorable propagation condition for Rossby waves
derived from 50 winters (1961–2010) in the Northern Hemisphere based on
different MVF values described in Fig. .
On the left the time-averaged zonal mean fields are used to
calculate the vertical wave number squared (only for (k,l)=(1,1)). On the
right the time mean of the vertical wave number is shown. It is clear that the
vertical wave number derived from the time-averaged zonal mean fields has less
noise than the time mean vertical wave number squared. We discuss this effect
in more detail in the manuscript. Theoretically there are various ways in
which one may reduce the level of noise in the time mean of the vertical
wave number. The advantage of our proposed method is that it maps well and in
a physical way on the list of criteria formulated in Table .
Alternatively one can use other statistical methods like truncated means or
trimmed means to reduce the noisiness.
Acknowledgements
Khalil Karami and Miriam Sinnhuber gratefully acknowledge funding by the
Helmholtz Society within the Helmholtz Young Investigators group: Solar
variability, climate, and the role of the mesosphere/lower thermosphere,
project NWG-642. NCEP Reanalysis data provided by the NOAA/OAR/ESRL PSD,
Boulder, Colorado, USA, from their Web site at
http://www.esrl.noaa.gov/psd.The article
processing charges for this open-access publication were
covered by a Research Centre of the Helmholtz
Association.Edited by: M. Heimann
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